Let μ be a complex measure on R n and f ∈ L 1 ( μ ) such that f ( x ) ≥ c &gt; 0 ( for some constant c &gt; 0 ) a.e. x ∈ R n . Then is it true that | ∫ R n f d μ | ≥ | ∫ R n c d μ | = c | μ ( R n ) | ? ? My first thinking process is to break each integral first in its real and imaginary part and then their corresponding positive and negative parts, i.e. ∫ R n f d μ = R e ( ∫ R n f d μ ) + i I m ( ∫ R n f d μ ) = R e + ( ∫ R n f d μ ) − R e − ( ∫ R n f d μ ) + i I m + ( ∫ R n f d μ ) − i I m − ( ∫ R n f d μ ) and then wanted to look at corresponding decompositions of the measure and estimate on each such segements. Also note that a complex measure by definition gives | μ ( R n ) | &lt; ∞ . Now how to proceed from here.Can someone please help?

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Let   μ be a complex measure on             R              n       and   f  ∈      L    1    (  μ  ) such that   f  (  x  )  ≥  c  &amp;gt;  0 ( for some constant   c  &amp;gt;  0 ) a.e.   x  ∈            R        n  . Then is it true that            |            ∫                            R                n              f  d  μ            |        ≥            |            ∫                            R                n              c  d  μ            |        =  c      |    μ  (            R        n    )      |     ? ? My first thinking process is to break each integral first in its real and imaginary part and then their corresponding positive and negative parts, i.e.      ∫                            R                n              f  d  μ  =  R  e            (            ∫                            R                n              f  d  μ            )        +  i  I  m            (            ∫                            R                n              f  d  μ            )        =  R      e    +              (            ∫                            R                n              f  d  μ            )        −  R      e    −              (            ∫                            R                n              f  d  μ            )        +  i  I      m    +              (            ∫                            R                n              f  d  μ            )        −  i  I      m    −              (            ∫                            R                n              f  d  μ            )      and then wanted to look at corresponding decompositions of the measure and estimate on each such segements. Also note that a complex measure by definition gives       |    μ  (            R        n    )      |    &amp;lt;  ∞ . Now how to proceed from here.Can someone please help?

marktje28 · Accepted Answer

You can pick   μ  =  2      δ    0    −      δ    1   and   f  (  0  )  =  1  ,  f  (  1  )  =  2. Then      ∫                            R                n              f  d  μ  =  2  f  (  0  )  −  f  (  1  )  =  2  ⋅  1  −  1  ⋅  2  =  0.However,  |  μ  (            R        n    )  |  =  |  2  −  1  |  =  1.This means the inequality is not even true for general signed measures.

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